The scalar curvature is the function \(S\) defined as the trace of the Ricci tensor :
\[ S={\tr}_g \Rc = {R_i}^i=g^{ij}R_{ij}. \]
\[
\DeclareMathOperator{\Rc}{Rc}
\DeclareMathOperator{\tr}{tr}
\]
The scalar curvature is the function \(S\) defined as the trace of the Ricci tensor :
\[ S={\tr}_g \Rc = {R_i}^i=g^{ij}R_{ij}. \]